Skip to main content
Chemistry LibreTexts

24.6: The Translational Partition Function of An Ideal Gas

  • Page ID
    151983
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    We can make use of Stirling’s approximation to write the translational contribution to \({ \ln Z_{IG}\ }\) per mole of ideal gas. This is

    \[ \ln \left[\frac{\left(z_t\right)^{\overline{N}}}{\overline{N}!}\right] =\overline{N} \ln z_t -\overline{N} \ln \overline{N} +\overline{N}=\overline{N}+\overline{N} \ln \frac{z_t}{\overline{N}} \nonumber \]

    (We omit the other factors in Stirling’s approximation. Their contribution to the thermodynamic values we calculate is less than the uncertainty introduced by the measurement errors in the molecular parameters we use.) In Section 24.3 we find the molecular partition function for translation:

    \[z_t= \left(\frac{2\pi mkT}{h^2}\right)^{3/2}V \nonumber \]

    For one mole of an ideal gas, \(\overline{V}={\overline{N}kT}/{P}\). The translational contribution to the partition function for one mole of an ideal gas becomes

    \[ \ln \left[\frac{\left(z_t\right)^{\overline{N}}}{\overline{N}!}\right] =\overline{N}+\overline{N} \ln \left[ \left(\frac{2\pi mkT}{h^2}\right)^{3/2}\frac{\overline{V}}{\overline{N}}\right] =\overline{N}+\overline{N} \ln \left[\left(\frac{2\pi mkT}{h^2}\right)^{3/2}\frac{kT}{P}\right] \nonumber \]


    This page titled 24.6: The Translational Partition Function of An Ideal Gas is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.